NAG Library Routine Document

S30JAF

S30JAF uses Merton's jump-diffusion model (Merton (1976)) to compute the price of a European option. This assumes that the asset price is described by a Brownian motion with drift, as in the Black–Scholes–Merton case, together with a compound Poisson process to model the jumps. The corresponding stochastic differential equation is,

\frac{d S}{S} = (α - λ k) d t + \hat{σ} {d W}_{t} + {d q}_{t} .

Here

α

is the instantaneous expected return on the asset price,

S

;

{\hat{σ}}^{2}

is the instantaneous variance of the return when the Poisson event does not occur;

{d W}_{t}

is a standard Brownian motion;

q_{t}

is the independent Poisson process and

k = E [Y - 1]

where

Y - 1

is the random variable change in the stock price if the Poisson event occurs and

E

is the expectation operator over the random variable

Y

This leads to the following price for a European option (see Haug (2007))

P_{call} = \sum_{j = 0}^{\infty} \frac{e^{- λ T} {(λ T)}^{j}}{j!} C_{j} (S, X, T, r, σ_{j}^{'}),

where

T

is the time to expiry;

X

is the strike price;

r

is the annual risk-free interest rate;

C_{j} (S, X, T, r, σ_{j}^{'})

is the Black–Scholes–Merton option pricing formula for a European call (see S30AAF).

\begin{array}{l} σ_{j}^{'} = \sqrt{z^{2} + δ^{2} (\frac{j}{T})}, \\ z^{2} = σ^{2} - λ δ^{2}, \\ δ^{2} = \frac{γ σ^{2}}{λ}, \end{array}

where

σ

is the total volatility including jumps;

λ

is the expected number of jumps given as an average per year;

γ

is the proportion of the total volatility due to jumps.

The value of a put is obtained by substituting the Black–Scholes–Merton put price for

C_{j} (S, X, T, r, σ_{j}^{'})

The option price

P_{i j} = P (X = X_{i}, T = T_{j})

is computed for each strike price in a set

X_{i}

i = 1, 2, \dots, m

, and for each expiry time in a set

T_{j}

j = 1, 2, \dots, n

4 References

Haug E G (2007) The Complete Guide to Option Pricing Formulas (2nd Edition) McGraw-Hill

Merton R C (1976) Option pricing when underlying stock returns are discontinuous Journal of Financial Economics 3 125–144

5 Parameters

1: $CALPUT$ – CHARACTER(1)Input

On entry: determines whether the option is a call or a put.

$CALPUT ='C'$: A call; the holder has a right to buy.
$CALPUT ='P'$: A put; the holder has a right to sell.

Constraint:

CALPUT ='C'

'P'

2: $M$ – INTEGERInput

On entry: the number of strike prices to be used.

Constraint:

M \geq 1

3: $N$ – INTEGERInput

On entry: the number of times to expiry to be used.

Constraint:

N \geq 1

4: $X (M)$ – REAL (KIND=nag_wp) arrayInput

On entry:

X (i)

must contain

X_{i}

, the

i

th strike price, for

i = 1, 2, \dots, M

Constraint:

X (i) \geq z ​ and ​ X (i) \leq 1 / z

, where

z = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, M

5: $S$ – REAL (KIND=nag_wp)Input

On entry:

S

, the price of the underlying asset.

Constraint:

S \geq z ​ and ​ S \leq 1.0 / z

, where

z = X02AMF ()

, the safe range parameter.

6: $T (N)$ – REAL (KIND=nag_wp) arrayInput

On entry:

T (i)

must contain

T_{i}

, the

i

th time, in years, to expiry, for

i = 1, 2, \dots, N

Constraint:

T (i) \geq z

, where

z = X02AMF ()

, the safe range parameter, for

i = 1, 2, \dots, N

7: $SIGMA$ – REAL (KIND=nag_wp)Input

On entry:

σ

, the annual total volatility, including jumps.

Constraint:

SIGMA > 0.0

8: $R$ – REAL (KIND=nag_wp)Input

On entry:

r

, the annual risk-free interest rate, continuously compounded. Note that a rate of 5% should be entered as 0.05.

Constraint:

R \geq 0.0

9: $LAMBDA$ – REAL (KIND=nag_wp)Input

On entry:

λ

, the number of expected jumps per year.

Constraint:

LAMBDA > 0.0

10: $JVOL$ – REAL (KIND=nag_wp)Input

On entry: the proportion of the total volatility associated with jumps.

Constraint:

0.0 \leq JVOL < 1.0

11: $P (LDP, N)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

P (i, j)

contains

P_{i j}

, the option price evaluated for the strike price

X_{i}

at expiry

T_{j}

for

i = 1, 2, \dots, M

and

j = 1, 2, \dots, N

12: $LDP$ – INTEGERInput

On entry: the first dimension of the array P as declared in the (sub)program from which S30JAF is called.

Constraint:

LDP \geq M

13: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $CALPUT = 〈value〉$ was an illegal value.

$IFAIL = 2$: On entry, $M = 〈value〉$ .
Constraint: $M \geq 1$ .

$IFAIL = 3$: On entry, $N = 〈value〉$ .
Constraint: $N \geq 1$ .

$IFAIL = 4$: On entry, $X (〈value〉) = 〈value〉$ .
Constraint: $X (i) \geq 〈value〉$ and $X (i) \leq 〈value〉$ .

$IFAIL = 5$: On entry, $S = 〈value〉$ .
Constraint: $S \geq 〈value〉$ and $S \leq 〈value〉$ .

$IFAIL = 6$: On entry, $T (〈value〉) = 〈value〉$ .
Constraint: $T (i) \geq 〈value〉$ .

$IFAIL = 7$: On entry, $SIGMA = 〈value〉$ .
Constraint: $SIGMA > 0.0$ .

$IFAIL = 8$: On entry, $R = 〈value〉$ .
Constraint: $R \geq 0.0$ .

$IFAIL = 9$: On entry, $LAMBDA = 〈value〉$ .
Constraint: $LAMBDA > 0.0$ .

$IFAIL = 10$: On entry, $JVOL = 〈value〉$ .
Constraint: $JVOL \geq 0.0$ and $JVOL < 1.0$ .

$IFAIL = 12$: On entry, $LDP = 〈value〉$ and $M = 〈value〉$ .
Constraint: $LDP \geq M$ .

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

The accuracy of the output is dependent on the accuracy of the cumulative Normal distribution function,

Φ

, occurring in

C_{j}

. This is evaluated using a rational Chebyshev expansion, chosen so that the maximum relative error in the expansion is of the order of the machine precision (see S15ABF and S15ADF). An accuracy close to machine precision can generally be expected.

8 Parallelism and Performance

S30JAF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

S30JAF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example computes the price of a European call with jumps. The time to expiry is

3

months, the stock price is

45

and the strike price is

55

. The number of jumps per year is

3

and the percentage of the total volatility due to jumps is

40 %

. The risk-free interest rate is

10 %

per year and the total volatility is

25 %

per year.

NAG Library Routine DocumentS30JAF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

S30JAF